Logarithm of Cumulative Distribution Function

Evaluate the natural logarithm of the cumulative distribution function for a Pareto (Type I) distribution.

The cumulative distribution function for a Pareto (Type I) random variable is

and zero otherwise. In the equation, alpha > 0 is the shape parameter and beta > 0 is the scale parameter.

Usage

var logcdf = require( '@stdlib/stats/base/dists/pareto-type1/logcdf' );

logcdf( x, alpha, beta )

Evaluates the natural logarithm of the cumulative distribution function (CDF) for a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var y = logcdf( 2.0, 1.0, 1.0 );
// returns ~-0.693

y = logcdf( 5.0, 2.0, 4.0 );
// returns ~-1.022

y = logcdf( 4.0, 2.0, 2.0 );
// returns ~-0.288

y = logcdf( 1.9, 2.0, 2.0 );
// returns -Infinity

y = logcdf( +Infinity, 4.0, 2.0 );
// returns 0.0

If provided NaN as any argument, the function returns NaN.

var y = logcdf( NaN, 1.0, 1.0 );
// returns NaN

y = logcdf( 0.0, NaN, 1.0 );
// returns NaN

y = logcdf( 0.0, 1.0, NaN );
// returns NaN

If provided alpha <= 0, the function returns NaN.

var y = logcdf( 2.0, -1.0, 0.5 );
// returns NaN

y = logcdf( 2.0, 0.0, 0.5 );
// returns NaN

If provided beta <= 0, the function returns NaN.

var y = logcdf( 2.0, 0.5, -1.0 );
// returns NaN

y = logcdf( 2.0, 0.5, 0.0 );
// returns NaN

logcdf.factory( alpha, beta )

Returns a function for evaluating the natural logarithm of the cumulative distribution function (CDF) of a Pareto (Type I) distribution with parameters alpha (shape parameter) and beta (scale parameter).

var mylogcdf = logcdf.factory( 10.0, 2.0 );
var y = mylogcdf( 3.0 );
// returns ~-0.017

y = mylogcdf( 2.5 );
// returns ~-0.114

Notes

In virtually all cases, using the logpdf or logcdf functions is preferable to manually computing the logarithm of the pdf or cdf, respectively, since the latter is prone to overflow and underflow.

Examples

var randu = require( '@stdlib/random/base/randu' );
var logcdf = require( '@stdlib/stats/base/dists/pareto-type1/logcdf' );

var alpha;
var beta;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu() * 8.0;
    alpha = randu() * 5.0;
    beta = randu() * 5.0;
    y = logcdf( x, alpha, beta );
    console.log( 'x: %d, α: %d, β: %d, ln(F(x;α,β)): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}